# Class 11 Maths NCERT Solutions for Chapter 13 Limits and Derivatives Exercise 13.1 ### Limits and Derivatives Exercise 13.1 Solutions

1. Evaluate the Given limit (x + 3)

Solution (x + 3) = 3 + 3 = 6

2. Evaluate the Given limit : (x - 22/7)

Solution 3. Evaluate the Given limit : πr2.
Solution πr2 = π(1)2 = π

4. Evaluate the Given limit : (4x + 3)/(x - 2)
Solution 5. Evaluate the Given limit : (x10 + x5 + 1)/(x – 1)
Solution 6. Evaluate the Given limit : [(x + 1)5 - 1]/x
Solution 7. Evaluate the Given limit : (3x2 - x - 10)/(x2 - 4)
Solution
At x = 2, the value of the given rational function takes the form 0/0 . 8. Evaluate the Given limit (x4 -81)/(2x2 - 5x - 3)
Solution
At x = 2, the value of the given rational function takes the form 0/0 . 9. Evaluate the Given limit : (ax + b)/(cx + 1)
Solution 10. Evaluate the Given limit : Solution At z = 1, the value of the given function takes the form 0/0 .
Put z1/6 = x so that z →1 as x →1. 11. Evaluate the Given limit (ax2 + bx + c)/(cx2 + bx + a) , a + b + c ≠ 0 .
Solution = 1 [a + b + c ≠ 0 ]

12. Evaluate the Given limit : Solution At x = -2, the value of the given function takes the form  0/0 . 13. Evaluate the Given limit sin ax/bx
Solution sin ax/bx
At x = 0, the value of the given function takes the form 0/0 . 14. Evaluate the Given limit : sin ax/sin bx, a,b ≠ 0

Solution (sin ax/sin bx), a,b ≠ 0
At x = 0, the value of the given function takes the form  0/0. 15. Evaluate the Given limit [sin(π -x)]/π(π - x)
Solution 16. Evaluate the given limit cos x/(π - x)
Solution 17. Evaluate the Given limit : (cos 2x - 1)/(cos x- 1)
Solution (cos 2x - 1)/(cos x- 1)
At x = 0, the value of the given function takes the form  0/0 .
Now, 18. Evaluate the Given limit (ax + x cos x)/b sin x
Solution (ax + x cos x)/b sin x
At x = 0, the value of the given function takes the form 0/0.
Now, 19. Evaluate the Given limit sec x
Solution 20. Evaluate the Given limit (sin ax + bx)/(ax + sin  bx) a,b, a+b ≠ 0.
Solution
At x = 0, the value of the given function takes the form 0/0 .
Now, 21. Evaluate the Given limit (cosec x - cot x)
Solution
At x = 0, the value of the given function takes the form ∞→∞.
Now, 22. Evaluate the Given limit Solution At x = π/2 , the value of the given function takes the form  0/0.
Now, put x - π/2 = y so that x → π/2, y → 0. 23. Find f(x)  and f(x) where Solution
The given function is  24. Find f(x) where Solution
The given function is 25. Evaluate f(x) where Solution It is observed that f(x) ≠ f(x).
Hence, f(x) does not exist.

26. Find f(x) where Solution
The given function is It is observed that f(x) ≠ f(x).
Hence, f(x) does not exist .

27. Find f(x) , where f(x) = |x| - 5
Solution
The given functions is f(x) = |x| - 5. 28. Suppose and if f(x) = f(1) what are possible values of a and b?
Solution
The given function is  f(x) = (a + bx) = a + b f(x) = (b - ax) = b - a
f(1) = 4
It is given that f(x) = f(1).
∴ f(x) = f(x) = f(x) = f(1)
⇒ a + b = 4 and b - a = 4
On solving these two equations, we obtain a = 0 and b = 4.
Thus, the respective possible values of a and b are 0 and 4.

29. Let a1, a2, . . ., an be fixed real numbers and define a function f ( x) = (x − a1) (x − a2)...(x − an). What is f(x) ? For some a ≠ a1, a2, ..., an, compute Solution
The given function is  f(x) = (x – a1)(x – a2) … (x - an). 30. If For what value (s)  of a does f(x) exists ?
Solution
The given function is When a = 0, 31. If the function f(x) satisfies [f(x) - 2]/[x2 -1] = π , evaluate f(x).
Solution 32. If For what integers m and n does f(x) and f(x) exist ?
Solution
The given function is 